Difference between revisions of "Pseudo-metric"
From Encyclopedia of Mathematics
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| − | ''on a set | + | ''on a set $X$'' |
| − | A non-negative real-valued function | + | A non-negative real-valued function $d$ defined on the set of all pairs of elements of $X$ (that is, on $X \times X$) and satisfying the following three conditions, called the axioms for a pseudo-metric: |
| − | a) if | + | a) if $x = y$, then $d(x,y) = 0$; |
| − | b) | + | b) $d(x,y) = d(y,x)$ (symmetry); |
| − | c) | + | c) $d(x,y) \le d(x,z) + d(z,y)$ (triangle inequality), where $x,y,z$ are arbitrary elements of $X$. |
| − | It is not required that | + | It is not required that $d(x,y) = 0$ implies $x=y$. A topology on $X$ is determined by a pseudo-metric $d$ on $X$ as follows: A point $x$ belongs to the closure of a set $A \subseteq X$ if $d(x,A) = 0$, where |
| + | $$ | ||
| + | d(x,A) = \inf_{a \in A} d(x,a) \ . | ||
| + | $$ | ||
| − | + | This topology is [[Completely-regular space|completely regular]] but is not necessarily [[Hausdorff space|Hausdorff]]: [[singleton]] sets can be non-closed. Every completely-regular topology can be given by a collection of pseudo-metrics as the lattice union of the corresponding pseudo-metric topologies. Analogously, families of pseudo-metrics can be used in defining, describing and investigating uniform structures. | |
| − | + | ====Comments==== | |
| + | See also [[Metric]], [[Quasi-metric]] and [[Symmetry on a set]]. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.L. Kelley, "General topology" , Springer (1975)</TD></TR>< | + | <table> |
| − | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> J.L. Kelley, "General topology" , Graduate Texts in Mathematics '''27''' Springer (1975) {{ISBN|0-387-90125-6}} {{ZBL|0306.54002}}</TD></TR> | |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Čech, "Topological spaces" , Interscience (1966) pp. 532 {{ZBL|0141.39401}}</TD></TR> | |
| − | + | </table> | |
| − | = | ||
| − | |||
| − | + | {{TEX|done}} | |
| − | |||
Latest revision as of 11:48, 23 November 2023
on a set $X$
A non-negative real-valued function $d$ defined on the set of all pairs of elements of $X$ (that is, on $X \times X$) and satisfying the following three conditions, called the axioms for a pseudo-metric:
a) if $x = y$, then $d(x,y) = 0$;
b) $d(x,y) = d(y,x)$ (symmetry);
c) $d(x,y) \le d(x,z) + d(z,y)$ (triangle inequality), where $x,y,z$ are arbitrary elements of $X$.
It is not required that $d(x,y) = 0$ implies $x=y$. A topology on $X$ is determined by a pseudo-metric $d$ on $X$ as follows: A point $x$ belongs to the closure of a set $A \subseteq X$ if $d(x,A) = 0$, where $$ d(x,A) = \inf_{a \in A} d(x,a) \ . $$
This topology is completely regular but is not necessarily Hausdorff: singleton sets can be non-closed. Every completely-regular topology can be given by a collection of pseudo-metrics as the lattice union of the corresponding pseudo-metric topologies. Analogously, families of pseudo-metrics can be used in defining, describing and investigating uniform structures.
Comments
See also Metric, Quasi-metric and Symmetry on a set.
References
| [1] | J.L. Kelley, "General topology" , Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002 |
| [a1] | E. Čech, "Topological spaces" , Interscience (1966) pp. 532 Zbl 0141.39401 |
How to Cite This Entry:
Pseudo-metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-metric&oldid=29453
Pseudo-metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-metric&oldid=29453
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article